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Ginzburg-Landau minimizers in perforated domains
with prescribed degrees
Leonid Berlyand, Petru Mironescu
To cite this version:
Ginzburg-Landau minimizers in perforated domains with
prescribed degrees
Leonid Berlyand(1), Petru Mironescu(2)
October 2004, with an update in June 2008
Abstract.
Suppose that Ω is a 2D domain with holes ω0, ω1, . . . , ωj, j = 1...k. In the perforated domain
A = Ω\ ( ∪k
j=0ωj) we consider the class J of complex valued maps having degrees 1 and −1 on
the boundaries ∂Ω, ∂ω0 respectively and degree 0 on the boundaries of other holes.
We investigate whether the minimum of the Ginzburg-Landau energy Eκ is attained in J , as
well as the asymptotic behavior of minimizers as the coherency length κ−1tends to 0. We show that
the answer to these questions is determined by the value of the H1-capacity cap(A) of the domain.
If cap(A) ≥ π (domain A is ”thin”), minimizers exist for each κ. Moreover they are vortexless and converge in H1(A) (and even better) to a minimizing S1-valued harmonic map as κ → ∞.
When cap(A) < π (domain A is ”thick”), we establish existence of quasi-minimizers (maps with “almost minimal energy”), which exhibit a different qualitative behavior : they have exactly two zeroes (vortices) rapidly converging to ∂A as κ → ∞ . Finally we formulate a conjecture on non-existence of the minimizers in thick domains.
Update. This preprint was written at the end 2004. We added (in June 2008) a short final section accounting subsequent developments.
1
Introduction
This study led to a natural question : what if the superconductor in the composite described below is not ideal (e.g., of type II) ?
Mathematically, this means that, in the above minimization problem, the Dirichlet integral for harmonic maps should be replaced by the Ginzburg-Landau (GL) functional. Then the existence question becomes highly nontrivial and it leads to the following problem
mκ = Inf Eκ(u) = 1 2 Z A |∇u|2+ κ 2 4 Z A (1− |u|2)2 ; u∈ J . (1.1)
Here, Eκ is a GL type energy (without magnetic field), A is a 2D perforated domain, i.e.
A = Ω\ ( ∪kj=0ωj), ωj ⊂ Ω, j = 0, . . . , k, ωj ∩ ωl=∅, j 6= l, (1.2)
with Ω, ωj, j = 0, . . . , k, simply connected bounded smooth domains. The classJ of testing maps
is
J = {u ∈ H1(A ; IR2); |u| = 1 a.e. on ∂A, deg(u, ∂Ω) = 1,
deg(u, ∂ω0) =−1, deg(u, ∂ωj) = 0, j = 1, . . . , k}. (1.3)
We thus consider a domain with finitely many fixed holes ωj. The constant κ−1 is the coherency
length (GL parameter). For the sake of our discussion, we will allow κ to be 0, so that throughout this paper we let κ≥ 0.
A point that needs clarification is whether the definition of J is meaningful. In other words, we discuss whether, given a map u ∈ H1(A) such that |u| = 1 a.e. on ∂A, we can define the
degree of u on each component of ∂A. For this purpose, we start by briefly recalling the definition and the basic properties of the degree of a continuous complex-valued map (see, e.g., [1]). Let Γ be a C1 simple closed curve in C| . Intuitively speaking, the degree of a map v 6= 0 is defined
as follows. Suppose that the image of Γ, v(Γ), is a closed curve that surrounds the origin. If we cover Γ once, then v(Γ) winds around the origin a number of times, either in the positive direction (counterclockwise), or in the negative direction (clockwise). Then the degree of v is the number of positive loops minus the number of negative loops around the origin. This integer depends on the sense we choose to cover Γ. To give the formal definition we assume Γ to be oriented, i.e. we consider a parametrization f : [0, 1] → Γ, with f(0) = f(1). Let v : Γ → C| be a continuous
map and set w = v◦ f. It is a simple exercise that, if v is always different from 0 on Γ, then we may write, on [0, 1], w(t) =|w(t)|eıϕ(t) for some continuous ϕ. Since w(0) = w(1), it follows that
ϕ(1)− ϕ(0) ∈ 2πZZ. The integer d = ϕ(1)− ϕ(0)
More generally, if a ∈ C| is not among the values of v, then we may define deg(v, Γ, a) as
deg(v− a, Γ, 0). It is easy to see that, if v has more regularity, say v ∈ C1, then
d = 1 2ıπ Z Γ 1 v ∂v ∂τ = 1 2ıπ Z Γ v |v|2 ∂v ∂τ, (1.4)
where τ is the tangent vector directly oriented with respect to the fixed orientation. In particular, if we change orientation on Γ, d changes to −d. A special case of the above formula is obtained when |v| = 1, i.e., when v ∈ C1(Γ; S1). In this case, we have
deg v = 1 2π Z Γ v∧ ∂v ∂τ, (1.5)
where we have used the following
Notation. If z = a + ıb, w = c + ıd∈ C| , then z∧ w = ad − bc. We will also use later the following
notation : if u, v are complex-valued maps, then u∧ ∇v =
u∧ ∂v ∂x u∧ ∂v ∂y .
We quote here the main properties of the degree of non vanishing maps : a) if
|v − w| < Min{|v(z)|, |w(z)| ; z ∈ Γ},
then deg v= deg w. In particular, the degree is continuous with respect to uniform convergence ; b) deg(vw) =deg v+deg w ; c) deg v = −deg v ; d) if a, b ∈ C| lie in the same connected
component of C| \ v(Γ), then deg (v, Γ, a) =deg (v, Γ, b) ; e) if v ∈ C1(S1; S1) has the Fourier
expansion v = X n∈ZZ aneınθ, then deg v = X n∈ZZ n|an|2. (1.6)
Assume now that v ∈ H1/2(Γ). Then the right-hand side of (1.6) makes sense ; so does the
right-hand side of (1.5) if we interpret it appropriately, i.e., if we write it as 1 2π < v1, ∂v2 ∂τ >H1/2(Γ),H−1/2(Γ) − < v2, ∂v1 ∂τ >H1/2(Γ),H−1/2(Γ) .
Definition 1 ([12]) Let v∈ H1/2(Γ; S1). Then deg v = 1 2π < v1, ∂v2 ∂τ >H1/2(Γ),H−1/2(Γ)− < v2, ∂v1 ∂τ >H1/2(Γ),H−1/2(Γ) . (1.7)
Warning. In order to keep the notations simple, we will write, in what follows, the degree formula in the form (1.5), even when v is only in H1/2.
The surprising feature of this degree of H1/2 maps is that it is still an integer ; this was proved
by L. Boutet de Monvel, see [12]. The degree is continuous with respect to H1/2 convergence (see
[20]), which is the extension of property a) to H1/2 maps. Properties b) and c) are still valid for
H1/2 maps ; see Section 2 and Appendix A. There is an analogue of property d), but it is more
delicate to state (and not used in this paper) ; we send the reader to [20] for details. On the other hand, (1.6) clearly holds when v ∈ H1/2, by the definition of the degreee. Finally, we note
that the definition of degree still depends on the orientation we choose on Γ !!since, if we change orientation, then ∂v
∂τ changes to − ∂v ∂τ.
One can, more generally, define the degree of a map v ∈ H1/2(Γ; C| ) provided its range is far
away from 0. More specifically, assume that there exist constants a, b > 0 such that a ≤ |v| ≤ b a.e. on Γ. Then we set
deg v = deg v
|v|. (1.8)
We now return to the definition of J . Let u ∈ H1(A) be such that |u| = 1 a.e. on ∂A and
set v =tr|∂Au. For each connected component Γ of ∂A, we have v∈ H1/2(Γ; S1) and thus we may
define the degree of v on Γ, provided we choose an orientation on Γ. Throughout this paper, we use the following convention : each component Γ of ∂A is oriented with the direct orientation with respect to A. The degrees we prescribe in the definition ofJ are the degrees of v computed with respect to this orientation. Thus, for example, if A = {z ; ρ < |z| < R} and u(z) = z/|z|, then deg(u, ∂ω0) = −1 and deg(u, ∂Ω) = 1. On the other hand, recall that each simply closed
planar curve Γ has a natural orientation (counterclockwise). With our convention, given a general domain A, the orientation of ∂Ω is the natural one, while the orientation of ∂ωj, j = 0, . . . , k, is
the opposite of the natural one.
We complete our discussion of degree by mentioning another basic property of the degree of continuous maps : f) Assume that u ∈ C(A; C| ) is such that u 6= 0 on ∂A. Assume also that
deg(u, ∂Ω) +
k X j=0
deg(u, ∂ωj)6= 0. (Here,the orientation on ∂A is direct with respect to A.) Then u
has has (at least) a zero in A. There is an analogue of f) for H1-maps, but the statement is more
We may now address a first natural question concerning the minimization problem (1.1)-(1.3)
Question 1. Is mκ attained ?
Before discussing this question, we start by recalling the most intensively studied minimization problem for the Ginzburg-Landau functional (see [10]), namely
eκ = Inf{Eκ(u) ; u ∈ L}, (1.9)
where
L = {u ∈ H1(G) ; tr∂Gu = g}. (1.10)
Here, G is a smooth bounded domain in IR2 and g ∈ H1/2(∂G; S1) is fixed. In this case, the
minimum is obviously attained in (1.9). The reason is that L is closed with respect to weak H1
convergence ; therefore, if we take a minimizing sequence for (1.9)-(1.10) that weakly converges to some u, this u is inL and clearly minimizes (1.9)-(1.10).
The situation is more delicate when we do not prescribe a Dirichlet boundary condition, but only degrees, as shown by the following
Example 1 Let
nκ = Inf{Eκ(u) ; u ∈ M}, (1.11)
where
M = {u ∈ H1(ID) ;|u| = 1 a.e. on S1, deg(u, S1) = 1}. (1.12) Here, ID is the unit disc and we consider the natural orientation on S1. Then, for each κ > 0,
nκ = π and nκ is not attained.
We will prove and extend this example in Section 4. In particular, this example implies that the classM is not closed with respect to weak H1 convergence (it is closed with respect to strong
H1 convergence since degree is continuous for the strong H1/2 convergence). Here is an example of sequence in M weakly converging in H1 to a map which is not in M :
Example 2 Let (an)⊂ (0, 1) be such that an → 1. Set un(z) =
z− an
1− anz
, z ∈ ID. Then un ⇀−1
weakly in H1.
Clearly un → −1 a.e. (weak H1 convergence will be established in Section 4). Example 2 is
adapted in Section 4 to the class J in order to prove the following
This implies that the existence of minimizers of (1.1)-(1.3) does not follow immediately from the direct method of the Calculus of Variations.
Before discussing Question 1 further, we mention some useful a priori bounds on mκ. Recall
that in the case of a prescribed Dirichlet data with non zero degree (thoroughly studied in [10]) the Ginzburg-Landau energy tends to infinity as κ→ ∞. However, a straightforward calculation shows that the energy remains bounded when we have only specified degrees on the boundary. More specifically, in Section 6 we prove that, for degrees 1, −1, 0, . . . , 0, we have
mκ ≤ 2π. (1.13)
Note that the right-hand side of (1.13) is independent of A and κ. In fact, our construction yields analogous upper bounds for arbitrary degrees.
Let us briefly sketch how this upper bound is obtained. We want to consider u≡ 1 as simplest testing map, however, this u has to be modified in order to satisfy the required degree conditions. The modified u can be described as follows :
(i) u equals 1 in the domain A\ (D ∪ ∆) (see Fig. 1 below) ;
(ii) on the boundaries of D and ∆, u has modulus 1 and degrees −1, 1 respectively (see Fig. 2) ; (iii) u is harmonic in D and in ∆.
By choosing appropriately the phases ϕ and ψ in Picture 2, we prove that Eκ(u)→ 2π as D and
∆ shrink to points.
Roughly speaking, such a testing function has a ”vortex” (zero of degree−1 or 1) in D and in ∆, and the energy of each vortex is almost π. We will call these functions ”vortex testing maps”.
There is yet another upper bound, which is obtained by considering all the possible testing maps of modulus 1 in A. More specifically, we consider the class
K = {u ∈ J ; |u| = 1 a.e. in A}. (1.14) It turns out thatK is not empty because the degrees we prescribe have total sum 0 ; see Lemma 2.2 below. It is known that, in K, the minimum of the Ginzburg-Landau energy is attained (see [10]). Let
I0 = Min {Eκ(u) ; u∈ K} = Min 1 2 Z A |∇u|2; u∈ K . (1.15) Then, clearly, mκ ≤ I0. (1.16)
A more delicate property (see Section 6) is
Figure 1: u equals 1 except on D and ∆
Clearly, (1.13) and (1.17) imply that mκ ≤ Min {I0, 2π}. This bound is close to optimal when
κ is large. Indeed, we prove in Sections 6 and 7 that, for any A, we have lim
κ→∞mκ = Min{I0, 2π}. (1.18)
It turns out that I0 can be expressed in explicit geometrical terms using Newtonian capacity
of the domain A ; this is done in Sections 2 and 3. Here is a very simple example that will be detailed and generalized in Section 2 :
Example 3 Let A ={z ; ρ < |z| < R}. Then the H1-capacity of A is cap(A) = 2π
ln(R/ρ) and
I0 =
2π2
cap(A). (1.19)
In Section 2, we show that (1.19) is valid for any domain of the form Ω\ ω0. Moreover, we
introduce a generalized capacity such that (1.19) still holds for an arbitrary perforated domain A = Ω\ (∪kj=0ωj). In Section 3, we provide yet another geometrical interpretation of I0 in terms
of conformal representations.
Formula (1.18) suggests that one has to distinguish between three types of domains : a) ”subcritical”, for which I0 < 2π (or, equivalently, cap(A) > π) ;
b) ”critical”, for which I0 = 2π (or cap(A) = π) ;
c) ”supercritical”, for which I0 > 2π (or cap(A) < π).
This terminology is motivated by our results concerning the existence and behavior of minimizers, that we discuss below. We illustrate a)-c) by using Example 3. When A is a circular annulus, a) corresponds to R/ρ < e2, b) to R/ρ = e2 and c) to R/ρ > e2. Intuitively, one should
think of subcritical domains as ”thin” domains, and of supercritical domains as ”thick” domains. This is obvious for a circular annulus. For a generic domain, this follows from the geometrical interpretation of H1-capacity.
We now return to the existence of minimizers. The main tool in proving existence is the following result, established in Section 5
Proposition 2 Assume that mκ < 2π. Then mκ is attained.
The first result of this type was established for the Yamabe problem by Th. Aubin in [3]. Such results subsequently proved to be extremely useful in minimization problems with possible lack of compactness of minimizing sequences ; see [19], [16], [17], [13] and the more recent papers [18] and [23].
Lemma 1 Let (un) be a bounded sequence in J . Assume that un⇀ u weakly in H1(A) (and thus
we must have |u| = 1 a.e. on ∂A). Then : lim inf 1 2 Z A |∇un|2 ≥ 1 2 Z A |∇u|2+π |1−deg(u, ∂Ω)|+|−1−deg(u, ∂ω0)|+ k X j=1 |deg(u, ∂ωj)| (1.20) and 1 2 Z A |∇u|2 ≥ π deg(u, ∂Ω) + k X j=0 deg(u, ∂ωj) . (1.21)
The proof of Lemma 1 is presented in Section 5. The argument there works for arbitrary fixed degrees instead of 1, −1, 0, . . . , 0. Intuitively, the estimate (1.20) shows that the minimal energy needed to jump from degree d (for the maps un) to degree δ (for u), on a component of ∂A, is
π|d − δ|. This ”price” is, in general, optimal as shown by the maps in Example 2.
As an immediate consequence of Proposition 2 and of the upper bound (1.17), we obtain the following
Theorem 1 Assume that A is subcritical or critical. Then mκ is attained for each κ≥ 0.
Remark 1 Minimizers of (1.1)-(1.3), whenever they exist, are smooth. This requires some proof, since minimizers or, more generally, critical points of Eκ in J , satisfy mixed type boundary
conditions : Dirichlet for the modulus, and Neumann for the phase. Smoothness of critical points is established in Appendix C. The discussion on the degree of H1/2 maps is not essential for the
understanding of our proofs. The main ideas can be understood by considering smooth maps in (1.1)-(1.3).
In the subcritical and critical cases, we further address the following natural Question 2. What is the behavior of minimizers uκ of (1.1)-(1.3) as κ→ ∞ ?
The answer is given by
Theorem 2 Assume that A is subcritical or critical. Let uκ be a minimizer of (1.1)-(1.3). Then,
up to a subsequence, uκ → u∞ in C1,α(A), ∀ 0 < α < 1. Here, u∞ is a minimizer of (1.14)-(1.15).
Remark 2 It is known that, for the GL equation, one can not improve the C1,α convergence to,
say C2 convergence (H. Brezis, personal communication).
We now turn to the supercritical case, i.e., we assume I0 > 2π. In this case, our analysis is less
complete, in particular, we were not able to determine whether the value mκ is attained or not in
J . However, a simple consequence of Proposition 2 is that, when A is fixed and κ varies from 0 to∞, there are only three possibilities
Theorem 3 Assume that A is supercritical. Then (exactly) one of the three following possibilities holds :
a) mκ is attained for all κ > 0 ;
b) mκ is never attained ;
c) there is some κ1 ∈ (0, ∞) such that : if κ < κ1, then mκ is attained, while if κ > κ1, then mκ
is not attained.
Which one of the possibilities a), b) or c) actually occurs for a given A remains at present an open question. However, when A is of the form Ω\ ω0, we were able to rule out possibility b) :
Proposition 3 Assume that A = Ω\ ω0. Then either a) or c) holds.
By a formal analysis, we believe that possibility a) never occurs, and we were thus led to the following
Conjecture. Assume that A is supercritical. Then there is a constant κ1 ≥ 0 such that, if κ > κ1,
then mκ is never attained.
Since we do not know whether, for large values of κ, there are minimizers of (1.1)-(1.3), we are led to consider ”quasi-minimizers”. These maps, which are defined at the beginning of Section 7, are solutions of the Ginzburg-Landau equation with almost minimal energy. The advantage of considering quasi-minimizers is that they do always exist, and that any minimizer of (1.1)-(1.3) (if it exists) is a quasi-minimizer. The relevant difference between the subcritcal/critical case and the supercritical case is that, in the supercritical case, quasi-minimizers develop ”vortices” for large values of κ.
The notion of a vortex is not clearly defined in the GL literature (although it is perfectly understood !). Here we discuss briefly the notion of a vortex for solutions of the 2D GL equation −∆uκ = κ2uκ(1− |uκ|2) in A. (1.22)
The role of this discussion is to clarify different possible meanings of a vortex, although we will not give its formal definition.
(i) The most common understanding is that a ”vortex of uκ” is a zero z of uκ. A bit more restrictive
definition requires in addition : a) that z is an isolated zero ; b) that the degree of uκ computed
Dirichlet boundary data that does not vanish on the boundary ; in this case, all zeroes of uκ are
isolated, see [24]. It is common belief that condition b) is not restrictive either, in the sense that, for large κ, zeroes of degree 0 cease to exist. This is not known in general, but it is proved in many situations ; in particular, this holds for minimizers of the Ginzburg-Landau energy with respect to a fixed boundary data of modulus 1, see [10].
Here, the vortices are defined intrinsically by the uκ’s. Note that, in this setting, the vortices are
not singularities, since each uκ is smooth.
(ii) There is however another perspective, when a vortex can be defined as a singularity of a map. Suppose that a given sequence uκ ∈ H1 converges (up to subsequences) to some map u at
least a.e. Then one could define a ”vortex of u” as a singularity of u (u is not smooth near this point). Note that, in this setting, one has to consider asymtotic behavior of the sequence uκ’s as
κ→ ∞, since it determines u.
It is a common belief that the vortices of u are related to the vortices of the uκ (defined above),
as follows : given a vortex z of u, there are, for large κ, vortices zκ of uκ (i.e., zeroes of uκ) such
that zκ → z. This property is not proved in all the possible situations, but it is known to hold in
many cases ; in particular, for a fixed boundary data ([10]). The converse is known to be false, i.e., vortices zκ of uκ need not approach a vortex of u ; for example, if the zκ’s ”escape to the
boundary”. Note, that while (i) describes vortices of smooth functions (solutions of GL PDE are smooth), (ii) introduces vortices of functions, which are not necessarily smooth.
(iii)
A different perspective is to start by considering “regular points” of (uκ). Suppose that (uκ) is
a family of functions in H1 such that u
κ → u strongly in H1 in some neighborhood of a point z.
Then z is called a regular point of the the family (uκ).
One expects that a point in A is a vortex of u (in the sense of (ii))) if and only if it is not a regular point of (uκ) ; this need not be true for points z on ∂A. This result was rigorously proved in
[10] for all the points in A, when the boundary data is fixed ; in other words, in that context, a point in A is regular if and only if it is not an accumulation point of vortices zk of the uκ’s. This
property is also known to hold, for points z ∈ A, in many other situations.
(iv) There is a fourth point of view, which is particularly useful when treating the Ginzburg-Landau equation in presence of the magnetic field or the 3D Ginzburg-Landau equation ; this point of view was first developed in [35] and [31]. Loosely speaking, a point z ∈ A is a ”concentration point (for (uκ))” if there is some C > 0 such that, for any neighborhood U of z, the energy of uκ
in U is at least C for large values of κ. The energy considered in this approach is usually the GL energy, possibly rescaled by an appropriate factor.
Concerning concentration points z ∈ A, the are two rigorous results one expects :
a) z is a concentration point (for (uκ)) if and only if z it is not a regular point (for (uκ) ;
Note that, for a point z ∈ A, one expects
z vortex⇐⇒ z limit of vortices of uκ ⇐⇒ z not a regular point ⇐⇒ z concentration point.
We may now state in an informal way the results we establish in Section 11 relative to the properties of the quasi-minimizers uκ for large values of κ in the supercritical case I0 > 2π :
a) uk has exactly two vortices, one of degree 1 near ∂A, the other one of degree −1 near ∂ω0 ;
b) any a.e. limit u of the uκ’s is vortexless. More precisely, it is a constant ;
c) up to subsequences, there are exactly two concentration points, one on ∂Ω, the other one on ∂ω0. All the other points of A are regular.
Note the contrast in between the subcritical/critical and the supercritical domains, when we consider the behavior of solutions for large values of κ : in the first case, minimizers do not vanish, by Theorem 2. With more work, one can prove that quasi-minimizers do not vanish neither. In the supercritical case, however, quasi-minimizers do have zeroes (and minimizers presumably cease to exist). In a different context (S2-valued harmonic maps with Dirichlet boundary conditions in a
circular annulus of radii R and ρ), the existence of a critical value of R/ρ determining a qualitative change in the behavior of minimizers was established in [8]. Similar split in behavior was described in physical context, see, e.g., [22].
Finally, we discuss uniqueness of minimizers. Note that, if the minimization problem (1.1)-(1.3) has a solution, then it has infinitely many, since whenever uκ is a minimizer, so is αuκ,∀ α ∈ S1.
Thus we can, at best, prove uniqueness modulo S1. In Section 10, we adapt the methods developed
in [21] and [31] and establish
Theorem 4 Assume that A is subcritical or critical. Then, for large κ, the minimizers of (1.1)-(1.3) are unique modulo multiplication with constants of modulus 1.
Nevertheless, our analysis does not include the result of [27] which asserts that, if A is a circular annulus of sufficiently small capacity, then the minimizers of (1.1)-(1.3) are unique for all κ. In this context, we mention the following natural question concerning circular annuli
Open Problem. Let A ={z ; ρ < |z| < R}. Assume that A is subcritical or critical. Is it true that, for all κ, the minimizers of (1.1)-(1.3) are unique modulo S1 ?
The paper is organized as follows
1. Introduction
2. Properties of the class K and of the degree 3. The geometrical interpretation of the capacity 4. Properties of the class J
5. On the existence of minimizers 6. Bounds for the minimal energy mκ
7. Properties of quasi-minimizers
8. Asymptotic behavior of the minimizers in the subcritical case I0 < 2π
9. Asymptotic behavior of the minimizers in the critical case I0 = 2π
10. ”Uniqueness” of the minimizers in the subcritical and critical case
11. Asymptotic behavior of the quasi-minimizers in the supercritical case I0 > 2π
12. Existence of stable critical points in the supercritical case Appendices
A. Degree of H1/2 maps and capacity
B. Zeroes of complex valued maps C. Smoothness of critical points
D. On the harmonic extension in a circular annulus E. Elementary estimates for conformal mappings 13. Update
2
Properties of the class
K and of the degree
2.1 The class K
We discuss here some properties of the class K defined in the Introduction. For later use, it will be of interest to consider, more generally, the class
K = KD, d0, . . . , dk ={u ∈ H
1(A; S1); deg(u, ∂Ω) = D, deg(u, ∂ω
j) = dj, j = 0, . . . , k}. (2.1)
The properties of K we present below are well-known to experts. However, since part of these results are not published yet, we will also present some proofs in Appendix A. The main references for this section are [10], [15] and [20].
Lemma 2.2. ([15]) We have K 6= ∅ ⇐⇒ D + k X j=0 dj = 0. (2.3)
Lemma 2.3. ([15]) Assume the compatibility condition
D +
k X j=0
dj = 0 (2.4)
satisfied. Let v∈ K be fixed. Then : a) we have
K ={veıϕ ; ϕ∈ H1(A; IR)}; (2.5) b) K is closed with respect to weak H1 convergence.
We now recall the main result in [10] concerning the class K : Lemma 2.4. ([10]) Assume (2.4) satisfied. Let
I = ID, d0, . . . , dk = Min 1 2 Z A |∇u|2 ; u∈ K. (2.6)
Then I is attained. Moreover :
a) the minimizer is unique up to a phase shift, i.e., if u, v are two minimizers of (2.6), then u = αv for some α∈ S1 ;
b) any minimizer is smooth ; c) we have I = 1 2 Z A |∇η|2. (2.7)
Here, η is smooth and it is the only minimizer of
Min 1 2 Z A |∇ζ|2+ 2πXk j=0 djζ|∂ωj; ζ ∈ L , (2.8) where
L ={ζ ∈ H1(A; IR); ζ = 0 on ∂Ω, ζ = const. on each ∂ωj, j = 0, . . . , k} ; (2.9)
d) if u is any minimizer of (2.6) and if η is as above, then
e) if u is any minimizer of (2.6), we may write locally (i.e., in simply connected sub domains of A) u = eıϕ, with ϕ smooth. The quantity X = ∇ϕ is globally defined, it can be computed as
X = u∧ ∇u and is the only solution of
div X = 0 in A XR · ν = 0 on ∂A ∂ωj X· τ = 2πdj, j = 0, . . . , k. . (2.11) Moreover, we have |X| = |∇u|; (2.12)
f ) the function η defined above is the only solution of
∆η = 0 in A η = 0 on ∂Ω η = Cj on ∂ωj, j = 0, . . . k Z ∂ωj ∂η ∂ν = 2πdj, j = 0, . . . , k . (2.13)
(Here, the constants Cj are a priori unknown and part of the problem.)
2.2 The class K
¿From now on, we specialize to the class K, i.e., we will always assume in what follows that D = 1, d0 =−1, dj = 0, j = 1, . . . , k. (2.14)
Note that K satisfies the compatibility condition (2.4), and thus Lemma 2.4 applies to K. In agreement with the notation used in the Introduction, we will write I0 instead of I1, −1, 0, . . . , 0.
The following properties of the function η introduced above will be useful later : Lemma 2.5. Assume (2.14) satisfied. Then :
a) 0 > Cj > C0, j = 1, . . . , k and 0 > η > C0 in A ;
b) if t ∈ (C0, 0) is not a critical value of η, then the level set {η = t} consists of a single simple
curve which encloses ∂ω0 and Z
{η=t}
|∇η| = 2π; (2.15)
c) I0 and C0 are related by
C0 =−
I0
In case of a circular annulus, we have explicit formulae for u, η, I0 and C0 :
Lemma 2.6. Assume that A ={z ; ρ < |z| < R}, i.e., Ω = {z ; |z| < R} and ω0 ={z ; |z| < ρ}.
Then :
a) all the minimizers of (1.14)-(1.15) are of the form u(z) = α z
|z| for some α∈ S 1 ; b) η(z) = ln|z| − ln R ; c) C0 = ln ρ− ln R ; d) I0 = π ln R ρ.
Corollary 2.1. When A is a circular domain, A = {z ; ρ < |z| < R}, the subcritical case corresponds to R/ρ < e2, the critical case to R/ρ = e2 and the supercritical case to R/ρ > e2.
We now turn to domains of the form A = Ω\ ω0. As we will see, in this case I0 and η are
related to the H1-capacity of A. We recall the definition of the H1-capacity of a hole in a 2D
domain (see, e.g., [32]) :
Definition 2.1. Let ω0, Ω be smooth bounded simply connected domains in IR2 such that ω0 ⊂ Ω.
Set A = Ω\ ω0. Then cap(A) = Min { Z A |∇v|2 ; v ∈ H1(A), v = 0 on ∂Ω, v = 1 on ∂ω 0}. (2.17)
Lemma 2.7. Assume A = Ω\ ω0. Then
I0 =
2π2
cap(A). (2.18)
Finally, we consider a general perforated domain A = Ω\ ∪k
j=0ωj. In this case, we introduce
the following analogue of the H1-capacity
Definition 2.2. The generalized H1-capacity of the domain A = Ω\ ∪k
j=0ωj is cap(A) = Min { Z A |∇v|2 ; v ∈ H1(A), v = 0 on ∂Ω, v = 1 on ∂ω 0, v = Dj on ∂ωj, j = 1, . . . , k}. (2.19)
In (2.19), the minimum is taken among all the v’s and all the constants Dj. Note that the hole
v of (2.19) is unique and satisfies ∆v = 0 in A v = 0 on ∂Ω v = 1 on ∂ω0 v = Dj on ∂ωj, j = 1, . . . k Z ∂ωj ∂v ∂ν = 0, j = 1, . . . , k . (2.20)
The proof of Lemma 2.7 (presented in Appendix A) combined with (2.20) yields immediately Lemma 2.8. Assume A = Ω\ ∪k j=0ωj. Then I0 = 2π2 cap(A). (2.21) 2.3 Symmetric domains
We end this section by considering symmetric domains. In this case, we prove that there are minimizers u0 of I0 that inherit the symmetry properties of the domain. Since the hole ω0 plays
a distinguished role, we have to start by providing a good notion of symmetric domains. Definition 2.3. Let O be an isometry of the plane. The domain A = Ω \ ∪k
j=0ωj is symmetric
with respect to O if
O(A) = A and O(ω0) = ω0. (2.22)
Note that, if A is not a circular annulus, we may assume thatO is either an orthogonal symmetry with respect to a line, or a O a rotation of angle 2π/n, n ≥ 2. Indeed, if O is a rotation of angle 2πθ, with θ 6∈ Q| , then A has to be a circular annulus. On the other hand, if O is a rotation of
angle 2πm/n, with (m, n) = 1, then an appropriate iteration of O is of a rotation of angle 2π/n and invariates A.
Lemma 2.9. Assume that A is O-symmetric. Then there is a minimizer u0 of I0 such that
u0(O(z)) = O(u0(z)), ∀ z ∈ A. (2.23)
3
The geometrical interpretation of the capacity
As we will see below, the capacity cap(A) is related to conformal representations. We recall some well known facts about conformal representations of multiply connected domains. We follow essentially [1]. To start with, consider the case A = Ω\ ω0. Recall that, in this case, A can be
R/ρ is uniquely determined by A. Indeed, circular annuli are conformally rigid, i.e., two annuli {z ; ρ1 <|z| < R1} and {z ; ρ2 <|z| < R2} are conformally equivalent if and only if R1/ρ1 = R2/ρ2
([1]). It turns out that the ratio R/ρ is related to cap(A) in a simple way and that the maps defined in Lemma 2.4, namely u and η, provide an explicit representation of A into{z ; ρ < |z| < R}. Definition 3.1. If u is a minimizer of (1.14)-(1.15) and η is the map defined in Lemma 2.4, let
f = fA, u = eηu. (3.1)
It is clear from Lemma 2.4 that f is holomorphic. Part of the following result is proved in [1] : Lemma 3.1. Assume that A = Ω\ ω0. Then :
a) if ρ, R are such that A can be conformally represented into {z ; ρ < |z| < R}, then R ρ = exp 2π cap(A) ; (3.2)
b) the map f is a conformal representation of A into the circular annulus
C = z ; exp − 2π cap(A) <|z| < 1 . (3.3)
c) f extends to a C1-diffeomorphism from A intoC such that f(∂Ω) = {z ; |z| = 1} and f(∂ω 0) = z ;|z| = exp − cap(A)2π
. Moreover, f preserves the natural orientation of simple curves.
Proof : Part a) follows from b) and the conformal rigidity of circular annuli. Part b) is proved in [1] except that the explicit formula for the small radius ofC obtained in [1] is ρ = eC0. But this ρ is
exactly the one given in b), thanks to Lemmas 2.6 and 2.8. We now turn to the proof of c). On the one hand, it is clear from the definition of f that f is smooth up to the boundary. Since|f| = eC0
on ∂ω0, we have f (∂ω0)⊂ z ;|z| = exp − cap(A)2π ; similarly, we have f (∂Ω)⊂ {z ; |z| = 1}. On ∂ω0, we have ∂ϕ ∂τ = ∂η ∂ν < 0 and Z ∂ω0 ∂ϕ ∂τ = Z ∂ω0 ∂η ∂ν =−2π.. (3.4)
Thus (with the natural orientations), f is an orientation preserving diffeomorphism from ∂ω0 into
z ;|z| = exp
− cap(A)2π
. Similar assertion holds for ∂Ω. Finally, f preserves the natural orientation of any simple curve in A, since it does so for ∂ω0.
We now turn to a general A = Ω\ ∪k
Definition 3.2. ([1]) A canonical slit region is a set E of the form E = {z ; ρ < |z| < R} \ ∪k
j=1Γj. (3.5)
Here, each Γj is a closed circular arc properly contained into some circle {z ; |z| = Rj}, ρ < Rj <
R, and these arcs are mutually disjoint.
We quote the following version of Lemma 3.1, essentially proved in [1] : Lemma 3.2. Assume A = Ω\ ∪k
j=0ωj. Then :
a) the map fA, u is a conformal representation of A into a canonical slit region C of radii R = 1, ρ = exp − 2π cap(A) ;
b) f extends to a C1-diffeomorphism from A∪ ∂Ω ∪ ∂ω
0 into C ∪ {z ; |z| = 1} ∪ z ;|z| = exp − 2π cap(A)
such that f (∂Ω) ={z ; |z| = 1} and f(∂ω0) = z ;|z| = exp − cap(A)2π . Moreover, f preserves the natural orientation of simple curves ;
c) Γj = f (∂ωj), j = 1, . . . , k ;
d) if A can be represented into a canonical slit region F of radii ρ < R through some conformal mapping h such that |h||∂ω0 <|h||∂Ω, then there are some α ∈ C| \ {0}, β ∈ S
1 such that F = αC and h = αβf . In particular, R ρ = exp 2π cap(A) . (3.6)
4
Properties of the class
J
4.1 On Example 1
We begin this section by discussing in detail the Example 1 mentioned in the Introduction. We will prove the following slightly more general fact
Lemma 4.1. Let U be a smooth bounded simply connected domain in C| . Set, for κ > 0,
nκ = Inf 1 2 Z U |∇v|2+ κ2 4 Z U
(1− |v|2)2 ; v ∈ H1(A; C| ), |v| = 1 a.e. on ∂U, deg (v, ∂U) = 1.
(4.1) Then :
a) nκ = π ;
Proof : Let v be a testing map in (4.1). Since |∇v|2 ≥ 2|Jac v| pointwise, we find that 1 2 Z U |∇v|2 ≥ Z U |Jac v| ≥ Z U Jac v = 1 2 Z ∂U v ∧ ∂v ∂τ = π deg (v, ∂U ) = π, (4.2)
by Lemma 2.1 and the degree formula (1.5). Thus nκ ≥ π. We claim that there is no testing map
v such that 1 2 Z U |∇v|2+ κ2 4 Z U (1− |v|2)2 = π. (4.3)
Indeed, by (4.2) this would require |v| = 1 a.e., so that v ∈ H1(U ; S1). However, in this case
Lemma 2.2 implies that deg (v, ∂U ) = 0, which is the desired contradiction. We complete the proof by showing that nκ = π. Since U is smooth, U can be conformally represented into the
unit disc ID through a map w which extends as a C1-diffeomorphism from U into ID. Moreover, since U is bounded, w preserves the natural orientations, so that we have deg (w, ∂U ) = 1 ; thus w is in the class of testing maps. Consider now, for a ∈ ID, α ∈ S1, the Moebius conformal
representations of ID into itself,
ua,α(z) = α
z− a
1− az, ∀ z ∈ ID, (4.4)
and let ua = ua,1. Set va= ua◦ w, which is again a testing map, since uapreserves the orientation
of S1. By using repeatedly conformality, we have
1 2 Z U |∇va|2 = Z U | Jac va| = Z U Jac va= area(va(U )) = π. (4.5)
On the other hand, if we consider a real, a∈ (0, 1), we claim that lim aր1 κ2 4 Z U (1− |va|2)2 = lim aր1 κ2 4 Z ID (1− |ua|2)2 Jac w = 0. (4.6)
Indeed, the last equality in (4.6) follows by dominated convergence, using the fact that, for each fixed z∈ ID, we have ua(z)→ −1 as a ր 1.
Remark 4.1. Here is another similar example : Let A = Ω\ ω0 and consider the class
J ={u ∈ H1(A; C| ) ; |u| = 1 on ∂A, deg(u, ∂Ω) = 1, deg(u, ∂ω
0) = 0}.
Then one may prove that, for each κ≥ 0, we have
Inf{Eκ(u) ; u∈ J} = π,
4.2 The class J
We now turn to the study of the class J . As in Section 2, we will consider, more generally, the class
J = JD, d0, . . . , dk
={u ∈ H1(A; C| );|u| = 1 a.e. on ∂A, deg(u, ∂Ω) = D, deg(u, ∂ωj) = dj, j = 0, . . . , k}. (4.7)
In contrast with Lemma 2.2 and Lemma 2.3 b), we have
Lemma 4.2. The class J is always nonempty and never closed with respect to weak H1
conver-gence.
Proof : Fix aj ∈ ωj, j = 0, . . . , k and a∈ A and let
v(z) = k Y j=0 z− a j |z − aj| −dj z− a |z − a| D+Pkj=0dj . (4.8)
Let g = v|∂A. Then any smooth extension of g to A is in J.
In order to prove the second property, let v be any smooth map in JD − 1, d0, . . . , dk. Let w be a
conformal representation of Ω into ID and let ua be the map defined by (4.4). Set va = ua◦ w :
Ω→ ID. We are going to modify va in order to obtain a map having modulus 1 on ∂A ; va does
not have this property, since we only have|va| = 1 on ∂Ω. We start by estimating |va| on ∪kj=0∂ωj.
Let K = w(∪kj=0ωj), which is a compact in ID. It is easy to see that there is some C > 0 such that
|ua(z)| ≥ 1 − C(1 − a), ∀ z ∈ K, ∀ a ∈ (1/2, 1), (4.9)
and thus
|va(z)| ≥ 1 − C(1 − a), ∀ z ∈ ∪kj=0∂ωj, ∀ a ∈ (1/2, 1). (4.10)
We define now the following family of maps Φt : C| → ID, 0 < t < 1/4 :
Φt(z) = z, if |z| ≤ 1 − 2t z |z|, if |z| ≥ 1 − t 2− 1− 2t |z| z, if 1− 2t ≤ |z| ≤ 1 − t ,
which clearly satisfies Φt(z) =
z
|z|, if |z| ≥ 1 − t, |∇Φt− ∇id| ≤ Ct if |z| ≤ 1, |∇Φt| ≤ C, (4.11) for some constant C independent of t. Let
By (4.10) and (4.11), wa ∈ J provided a is sufficiently close to 1. Moreover, we have wa → −v 6∈ J
a.e. as a ր 1. This proves the lack of weak closedness of J provided we establish that (wa) is
bounded in H1(A). This is clear since on the one hand we have |w
a| ≤ |v| ≤ C and on the other
hand we have Z A |∇wa|2 ≤ 2 Z A (|v|2|∇(Φ√ 1−a◦ va)|2 +|Φ√1−a◦ va|2|∇v|2)≤ C Z A (|∇va|2+|∇v|2)≤ C, (4.12) by (4.5).
We next establish a lower bound for maps in J that will be useful later. Lemma 4.3. Let u∈ J. Then
1 2 Z A |∇u|2 ≥ π D + k X j=0 dj . (4.13)
Proof : We have, by Lemma 2.1 and the degree formula (1.5), 1 2 Z A |∇u|2 ≥Z A |Jac u| ≥ Z A Jac u = 1 2 Z ∂A u∧ ∂u ∂τ = π D + k X j=0 dj . (4.14)
4.3 Smoothness of critical points
We state here the following regularity result, whose proof is presented in Appendix C
Lemma 4.4. Let vκ ∈ J be a critical point of the Ginzburg-Landau energy Eκ with respect to J .
Then :
a) vκ ∈ C∞(A). In particular, near ∂A we may locally write vκ = ρeıψ ;
b) vκ satisfies the system −∆vκ = κ2vκ(1− |vκ|2) in A ρ = 1 on ∂A ∂ψ ∂ν = 0 on ∂A ; (4.15) c) |vκ| ≤ 1 in A.
Remark 4.2. Near ∂A, the (local) phase ψ of vκ is not unique. However, ∇ψ is uniquely
deter-mined and may be computed as vκ |vκ|∧ ∇
v κ
|vκ|
Remark 4.3. For later use, we mention that, if in the set {z ; vκ 6= 0}, we write locally (i.e., on
simply connected domains V ), vκ = ρκeıψκ = ρeıψ, then ρ and ψ satisfy ( −∆ρ = κ2ρ(1− ρ2)− ρ|∇ψ|2 in V ρ = 1 on ∂A∩ V (4.16) and respectively ( −div(ρ2∇ψ) = 0 in V ν· ∇ψ = 0 on ∂A ∩ V . (4.17)
5
On the existence of minimizers
5.1 A sufficient condition for the existence of minimizers This part is devoted to the proof of the following
Proposition 5.1. Assume that mκ < 2π. Then mκ is attained.
Proof : Let (un) be a minimizing sequence for EκinJ and let u be such that, up to a subsequence
un ⇀ u weakly in H1(A). Set gn = tr∂Aun, g = tr∂Au, so that gn⇀ g weakly in H1/2(∂A). Since
H1/2(∂A) is compactly embedded into L2(∂A), we have g
n → g in L2(∂A). In particular, up to
some further subsequence, we may assume that gn→ g a.e. Thus |g| = 1 a.e. on ∂A, and therefore
u∈ JD, d
0, . . . , dk for some integers D, d0, . . . , dk. By the Fatou lemma, we have
Eκ(u)≤ lim
n→∞Eκ(un) = mκ. (5.1)
Therefore, it suffices to prove that D = 1, d0 =−1, dj = 0, j = 1, . . . , k, i.e., that u∈ J .
We have 1 2 Z A |∇un|2 = 1 2 Z A |∇((un− u) + u)|2 = 1 2 Z A |∇(un− u)|2+ 1 2 Z A |∇u|2+ Z A ∇(un− u) · ∇u, (5.2) which implies 1 2 Z A |∇un|2 = 1 2 Z A |∇(un− u)|2+ 1 2 Z A |∇u|2+ o(1) as n→ ∞, (5.3)
since un− u ⇀ 0 weakly in H1(A).
Let vn, v be the harmonic extensions of gn, g respectively. Using the fact that gn ⇀ g weakly in
H1/2(∂A), we find that v
n→ v in Cloc1 (A), by standard elliptic estimates ([26]). Consider smooth
bounded disjoint neighborhoods of the ωj’s, U0, . . . , Uk, such that
and let U be a smooth domain such that ∪k j=0Uj ⊂ U ⊂ U ⊂ Ω. (5.5) Since 1 2 Z A |∇(un− u)|2 ≥ 1 2 Z A |∇(vn− v)|2 ≥ Z Ω\U |∇(vn− v)|2+ k X j=0 Z Uj\ωj |∇(vn− v)|2, (5.6)
we find, using the pointwise inequality |∇(vn− v)|2 ≥ 2|Jac (vn− v)| and Lemma 2.1, that
1 2 Z A |∇(un− u)|2 ≥ 1 2 Z ∂(Ω\U) (vn− v) ∧ ∂(vn− v) ∂τ + 1 2 k X j=0 Z ∂(Uj\ωj) (vn− v) ∧ ∂(vn− v) ∂τ . (5.7)
Recalling that vn→ v in Cloc1 (A), we obtain Z ∂(Ω\U) (vn− v) ∧ ∂(vn− v) ∂τ = Z ∂Ω (vn− v) ∧ ∂(vn− v) ∂τ + o(1), as n→ ∞ (5.8) and Z ∂(Uj\ωj) (vn− v) ∧ ∂(vn− v) ∂τ = Z ∂ωj (vn− v) ∧ ∂(vn− v) ∂τ + o(1), j = 1, . . . , k, as n→ ∞. (5.9) Since un and vn (respectively u and v) agree on ∂A we have, by the degree formula (1.5),
Z ∂Ω vn∧ ∂vn ∂τ = 2π, Z ∂Ω v∧ ∂v ∂τ = 2πD, (5.10) Z ∂ω0 vn∧ ∂vn ∂τ =−2π, Z ∂ω0 v ∧ ∂v ∂τ = 2πd0 (5.11) and Z ∂ωj vn∧ ∂vn ∂τ = 0, Z ∂ωj v∧ ∂v ∂τ = 2πdj, j = 1, . . . , k. (5.12)
Finally, we use the fact that Z Γ v∧ ∂vn ∂τ =− Z Γ ∂v ∂τ ∧ vn= Z Γ vn∧ ∂v ∂τ. (5.14)
The above equality is clear, using integration by parts, when both vn and v are smooth. The
general is obtained through approximation with smooth functions. By combining (5.3) with (5.7)-(5.14), we find 2π > mκ ≥ lim inf 1 2 Z A |∇un|2 ≥ π |D − 1| + |d0+ 1| + k X j=1 |dj| + 1 2 Z A |∇u|2, (5.15) which proves the Price Lemma (Lemma 1). Using the lower bound provided by Lemma 4.3, we are finally led to
2π > π |D − 1| + |d0+ 1| + k X j=1 |dj| + π D + k X j=0 dj ≡ πM + πN. (5.16)
We claim that the right-hand side of (5.16) is ≥ 2π unless
D = 1, d0 =−1, dj = 0, j = 1, . . . , k ; (5.17)
in other words, that (5.16) implies that u∈ J .
Indeed, if (5.17) does not hold, then : either exactly one of the equalities in (5.17) is violated, and thus D +
k X j=0
dj 6= 0, and the conclusion is clear, since M ≥ 1 and N ≥ 1 ; or, at least two of the
inequalities in (5.17) are false, and then the conclusion is again clear, since M ≥ 2. Therefore, u∈ J and the proof of the proposition is complete.
If we examine the above proof, we see that it has as a byproduct the following
Corollary 5.1. Assume that (un) ⊂ J is such that un ⇀ u weakly in H1(A) to some u 6∈ J .
Then lim inf n→∞ 1 2 Z A |∇un|2 ≥ max 2π , π + 1 2 Z A |∇u|2 ≥ 2π. (5.18)
Corollary 5.2. Assume that I0 < 2π. Then mκ is attained for each κ > 0.
Proof : Since any minimizer u of (1.14)-(1.15) belongs to J , we have
Corollary 5.3. Assume that I0 = 2π. Then mκ is attained for each κ > 0.
Proof : Let u be a minimizer of (1.14)-(1.15)and g be the restriction of u to ∂A. Let w attain the minimum of Eκ in the following subclass of J :
L = {v ∈ H1(A; C| ); tr
∂Av = g}. (5.20)
The minimum is clearly attained, and any minimizer w belongs to J and satisfies the Ginzburg-Landau equation
−∆w = κ2w(1− |w|2). (5.21)
We claim that u (which belongs to L) is not a minimizer of Eκ inL. Indeed, otherwise we would
have, by (5.21) and the fact that|u| = 1, that ∆u = 0. Using once again the fact that |u| = 1, we find that u is a constant. This contradicts the fact that u∈ K. In conclusion,
Eκ(w) < Eκ(u) = I0 = 2π, (5.22)
so that mκ < 2π and the conclusion follows from Proposition 5.1.
Corollary 5.4. Assume that mκ′ < 2π for some κ′. Then there is some κ′′ > κ′ such that mκ is
attained if k′ < κ < κ′′.
Proof : Let v∈ J be such that Eκ′(v) < 2π. Then clearly Eκ(v) < 2π if κ is sufficiently close to
κ′, and we conclude by applying Proposition 5.1 to any such κ.
5.2 Existence of minimizers for small κ when A = Ω\ ω0
Throughout this part, we assume that A is an annular domain, i.e., that A = Ω\ ω0. In this
case, we prove that, for small values of κ, there is a minimizer of (1.1)-(1.3). Moreover, we will determine the exact value of m0, as well as all the minimizers of (1.1)-(1.3). We start with the
following
Lemma 5.1. Assume that A = Ω\ ω0. Then m0 < 2π.
By combining Lemma 5.1 and Corollary 5.4 we obtain the following
Corollary 5.5. Assume that A = Ω\ ω0. Then there is some κ1 > 0 such that mκ is attained for
0≤ κ < κ1.
Proof of Lemma 5.1 : We could obtain Lemma 5.1 directly from Proposition 5.2. However, we feel that the argument below, which provides, for a given harmonic map, the harmonic extension of its trace, has its own interest.
Let u be a fixed minimizer of (1.14)-(1.15) and let η be as in Lemma 2.4. Let f be a smooth real function to be determined later and set u0 = f (η)u. Then
by Lemma 2.4. By the coarea formula (see, e.g., [25]) and (5.23), we have 1 2 Z A |∇u0|2 = 1 2 Z IR ( Z {η=t} (f′2(t) + f2(t))|∇η|dl)dt = π Z IR (f′2(t) + f2(t))dt ; (5.24)
the last equality in (5.24) follows from Lemma 2.5. Recall that, by Lemma 2.4, η is constant on ∂Ω and on ∂ω0 ; more specifically, by Lemma 2.7 we have η = 0 on ∂Ω and η = C0 on ∂ω0, where
C0 = −
I0
π. Assuming now that f (0) = f (C0) = 1, we obtain u0 ∈ J . If f(0) = f(C0) = 1, the right-hand side of (5.24) is minimal for f (t) = aet + be−t, where a, b satisfy a + b = 1 and
aeC0 + be−C0 = 1. Substituting this f into (5.24) yields
m0 ≤ 1 2 Z A |∇u0|2 = 2π 1− eC0 1 + eC0 = 2π 1− e−I0/π 1 + e−I0/π < 2π. (5.25)
In particular, for a circular annulus A = {z ; ρ < |z| < R}, we obtain, with the help of Lemma 2.6, that
m0 ≤ 2π
R− ρ
R + ρ. (5.26)
Remark 5.1. It is easy to see that the map u0 we constructed above may be also obtained in the
following way : let u be a minimizer of (1.14)-(1.15) and let g be its restriction to ∂A. Then u0 is
the harmonic extension of g to A. Recall that, by Lemma 2.4, the minimizers u of I0 are unique
up to a phase shift. Therefore, the maps u0 constructed above are unique up to a phase shift. We
will see below that the above construction is optimal, in the sense that the above maps u0 are
precisely the minimizers of (1.1)-(1.3) for κ = 0 and that ”≤” in (5.25) is actually ”=”. Remark 5.2. If we repeat the above construction when A = Ω\ (∪k
j=0ωj) with k ≥ 1, in general
we obtain 1 2
Z
A
|∇u0|2 > 2π ; this can be proved by considering appropriate A’s. Thus, when k ≥ 1,
we can not derive from the above construction that m0 is attained.
We next prove that the above construction is optimal. Proposition 5.2. Assume that A = Ω\ ω0. Then :
a) m0 = 2π
1− eC0
1 + eC0 ;
b) the minimizers of (1.1)-(1.3) for κ = 0 are precisely the maps u0 = f (η)u, where f (t) =
aet+ be−t, a + b = 1 and aeC0 + be−C0 = 1 and u is a minimizer of (1.14)-(1.15).
Proof : We start with the case where A is a circular annulus, A ={z ; ρ < |z| < R}. Recall that, by Lemma 2.6, in this case we have u(z) = α z
A of the restriction of u to ∂A. Therefore, in polar coordinates, we have
u = αr
2 + Rρ
r(R + ρ)e
ıθ. (5.27)
By Lemma D.3, the maps given by (5.27) are precisely the minimizers of (1.1)-(1.3) for κ = 0 and a) holds for this A. In particular, in this case the minimizers of (1.1)-(1.3) for κ = 0 are unique up to a phase shift.
We now turn to a general A = Ω\ ω0. Recall that, by Lemma 3.1 c), there is a conformal
representation F from A into C that extends to a C1 orientation preserving diffeomorphism from
A into C that preserves the natural orientations of curves ; here, C = {z ; ρ < |z| < R} and R
ρ = e
I0/π
. Thus, with obvious notations, we find that
J (A) ∋ v 7→ v ◦ F−1 ∈ J (C) (5.28)
is a bijection. Moreover, since F is a conformal representation, we have 1 2 Z A |∇v|2 = 1 2 Z C |∇(v ◦ F−1)|2. (5.29) In particular, using obvious notations, we find that
m0(A) = m0(C) = 2π R− ρ R + ρ == 2π 1− (R/ρ)−1 1 + (R/ρ)−1 = 2π 1− e−I0/π 1 + e−I0/π, (5.30)
the last equality following from Lemma 3.1.
Therefore, the maps constructed in Lemma 5.1 are, in A, minimizers of (1.1)-(1.3) for κ = 0. Moreover, since in C the minimizers of (1.1)-(1.3) for κ = 0 are determined up to a phase shift and so are the maps constructed in Lemma 5.1, it follows that these maps are all the minimizers in A of (1.1)-(1.3) for κ = 0.
6
Bounds for the minimal energy m
κ6.1 Upper bounds for mκ
We start by constructing appropriate test functions needed in order to derive sharp upper bounds for mκ.
Lemma 6.1. There is a sequence (un)⊂ J1, 0, . . . , 0 such that :
Proof : We make use of the functions considered in the proof of Lemma 4.1. Let w be a conformal representation of Ω into ID that extends smoothly up to the boundary, let (an)⊂ (0, 1)
be a sequence such that an → 1 and set vn = uan ◦ w. We next correct the map vn in order to
have modulus 1 on the boundary. To this purpose, we start by noting that |vn+ 1| ≤ C(1 − an)
on ∂A\ ∂Ω. With Φt as in the proof of Lemma 4.2, we set un = Φ√1−an ◦ vn in A. Then this un
belongs to J1, 0, . . . , 0 for sufficiently large n. Properties a) and b) are clear for this choice of un,
by construction. Moreover, if we extend un (the extension being still denoted by un) to Ω by the
same formula, properties a) and b) still hold with A replaced by Ω. On the other hand, we clearly have, by the property (4.11) of Φt, that
|∇un− ∇vn| ≤ C √ 1− an|∇vn| in Ω. (6.1) Therefore, lim n→∞ Z A |∇un|2 = lim n→∞ Z Ω |∇un|2 = lim n→∞ Z Ω |∇vn|2 = 2 Z Ω |Jac vn| = 2 Z Ω
Jac vn= 2 area (ID) = 2π,
(6.2) since vn is a conformal representation of Ω into ID.
Similarly, we have
Lemma 6.2. There is a sequence (vn)⊂ J0, −1, 0, . . . , 0 such that :
a) |vn| ≤ 1 in A, ∀ n; b) vn→ −1 in Cloc(A \ ∂ω1 0) ; c) lim n→∞ Z A |∇vn|2 = 2π.
Proof : We may assume that 0 ∈ ω0. Let g(z) = 1/z and B = g(A). Then B = O\ ∪kj=0Uj,
where O is the domain enclosed by g(∂ω0), U0 the domain enclosed by g(∂Ω), and Uj the domain
enclosed by g(∂ωj), j = 1, . . . , k. Construct, in B, a sequence (un) as in Lemma 6.1 and let
vn = un◦ g = un◦ g−1. Clearly, (vn) has the properties a) and b). As for c), it follows from Lemma
6.1 c) and the fact that
Z A |∇vn|2 = Z A |∇(un◦ g)|2 = Z B |∇un|2 = Z B |∇un|2, (6.3)
since g is a conformal representation.
Remark 6.1. In Section 11, we will give a geometrical interpretation of the maps un and vn
constructed above.
Proposition 6.1. We have, for each κ≥ 0 and each A, the following upper bounds
mκ ≤ 2π (6.4)
and
mκ < I0. (6.5)
Proof : Inequality (6.5) follows from the proof of Corollary 5.3, where we establish that, with u a minimizer of (1.14)-(1.15), there is some w∈ J such that Eκ(w) < Eκ(u) = I0.
For the proof of (6.4), let wn = unvn, where un, vn are given by the two preceding lemmas. By
(2.13), we have wn ∈ J . On the one hand, we have
lim
n→∞ Z
A
(1− |wn|2)2 = 0, (6.6)
by dominated convergence, thanks to a) and b) in Lemma 6.1 and Lemma 6.2. On the other hand, let U , V be smooth open sets such that
U, V ⊂ A, U ∩ V = ∅, U ∪ V = A, ∂Ω ⊂ U, ∂ω0 ⊂ V .
Then, by Lemma 6.1. and Lemma 6.2, we find
Z U |un∇vn+ vn∇un|2 = Z U |∇un|2+ o(1), Z V |un∇vn+ vn∇un|2 = Z V |∇vn|2+ o(1) as n→ ∞, (6.7) so that Z A |∇wn|2 = Z A |un∇vn+ vn∇un|2 = Z U |∇un|2+ Z V |∇vn|2+ o(1) = 4π + o(1) as n→ ∞ ; (6.8)
for the last equality, we use again Lemma 6.1 b) and Lemma 6.2 b). By combining (6.6) and (6.8), we find that
lim
n→∞Eκ(wn) = 2π, (6.9)
and (6.4) follows.
Corollary 6.1. Assume that mκ′ is attained for some κ′ > 0. Then mκ is attained for 0≤ κ < κ′.
Proof : Let uκ′ be a minimizer of (1.1)-(1.3) for κ = κ′. Since uκ′ satisfies
−∆uκ′ = κ′2uκ′(1− |uκ′|2), (6.10)
we noticed, during the proof of Corollary 5.3, that we can not have|uκ′| = 1. Therefore,
Z
A
and thus
Eκ(uκ′) < Eκ′(uκ′)≤ 2π if 0≤ κ < κ′. (6.12)
The conclusion follows now from Proposition 5.1.
Corollary 6.2. Assume that mκ′ is not attained for some κ′ ≥ 0. Then mκ′ = 2π and mκ is not
attained for κ > κ′.
Corollary 6.3. For each A, there is some κ′ ∈ [0, ∞] such that :
a) mκ is always attained for 0≤ κ < κ′;
b) mκ is never attained for κ > κ′.
If, in addition, κ′ <∞, then m
κ′ = 2π and mκ = 2π for κ > κ′.
Remark 6.2. When A has a single hole, it follows from Proposition 5.2. that κ′ > 0. We do not
know whether, when A has more than one hole, we always have κ′ > 0.
Remark 6.3. The conjecture mentioned in the introduction is equivalent to: I0 > 2π =⇒ κ′ <∞.
6.2 Asymptotic behavior of mκ
Among other facts, we prove below that the upper bounds established in Proposition 6.1 are asymptotically optimal as κ→ ∞.
Corollary 6.4. Assume that I0 < 2π. Then :
a) lim
κ→∞mκ = I0;
b) up to subsequences, minimizers uκ → u strongly in H1(A) as κ → ∞, where u is some minimizer
of (1.14)-(1.15).
Proof : First note that mκ is non-decreasing with κ, so that the limit in a) exists. Let u be such
that, along some subsequence, uκn ⇀ u weakly in H
1(A) and a.e. Since
1 2
Z
A
|∇u|2 ≤ lim infn→∞ 1 2
Z
A
|∇uκn|
2
≤ lim infn→∞ Eκn(uκn) = lim
κ→∞Eκ(uκ)≤ I0 < 2π, (6.13)
we find by Corollary 5.1 that u∈ J and that 1 2
Z
A
|∇u|2 ≤ I0. (6.14)
On the other hand, Z
A
(1− |uκ|2)2 ≤
4
so that |u| = 1 a.e. Therefore, u ∈ K, so that u has to be a minimizer of (1.14)-(1.15), by (6.14). Recalling (6.13) , we find that
lim n→∞ Z A |∇uκn| 2 =Z A |∇u|2, (6.16)
and thus uκn → u strongly in H
1(A). Part a) follows from (6.13) and (6.16).
Corollary 6.5. Assume that I0 = 2π. Then lim
κ→∞mκ = I0 = 2π.
Proof : Let u be such that, up to some subsequence, uκn ⇀ u weakly in H
1(A) and a.e. If u∈ J ,
then u is a minimizer of (1.14)-(1.15) and the conclusion follows as in the previous Corollary. If u6∈ J, by Corollary 5.1 we find that
2π≥ lim
κ→∞Eκ(uκ) = lim infn→∞ Eκn(uκn)≥ lim infn→∞
1 2 Z A |∇uκn| 2 ≥ 2π, (6.17) and the conclusion follows again.
Corollary 6.6. Assume that I0 > 2π. Then lim
κ→∞mκ = 2π.
Proof : Due to the corrolaty 6.3 if κ′ <∞ then conlsuion follows. Thus we assume that κ′ =∞
so mk is attained and we can consider a sequence of minimizers uκn” Let u be such that, up to
some subsequence, uκn ⇀ u weakly in H
1(A) and a.e. Since
1 2
Z
A
|∇u|2 ≤ lim inf n→∞ 1 2 Z A |∇uκn| 2 ≤ lim inf n→∞ Eκn(uκn) = limκ→∞Eκ(uκ)≤ 2π, (6.18)
we cannot have u∈ J . For otherwise, as in the proof of Corollary 6.4, we would have u ∈ K and 2π < I0 ≤ 1 2 Z A |∇u|2 ≤ 2π, (6.19)
which is impossible. Thus u 6∈ J and the conclusion follows by combining (6.18) with Corollary 5.1.
We will need later the following refinement of (6.5)
Lemma 6.3. There are some constants C = C(A) > 0 and κ′ = κ′(A) > 0 such that
mκ ≤ I0−
C
Proof : Let u be a minimizer of (1.14)-(1.15) and g =tr∂Au. Let, for κ > 0, vκ ∈ J be the
minimizer of Eκ in the class L given by (5.20). We claim that
vκ → u strongly in H1(A) as κ→ ∞. (6.21)
Indeed, let v be any possible weak limit of some subsequence of (uκ). Since Eκ(vκ)≤ Eκ(u) = I0,
we find as in the proof of Corollary 6.4 that |v| = 1 and 1 2
Z
A
|∇v|2 ≤ I0. On the other hand, we
have tr∂Au=tr∂Av, so that v ∈ K. Therefore, v is a minimizer of (1.14)-(1.15). By Lemma 2.4,
there is some α∈ S1 such that v = αu. This α has to be 1, since u and v agree on ∂A. Finally,
we proceed as in the proof of Corollary 6.4 to obtain the strong H1 convergence claimed in (6.21).
We now invoke the following result in [9] Lemma 6.4. ([9]) We have
κ2(1− |vκ|2)→ |∇u|2 in Cloc(A) as κ→ ∞ (6.22)
and lim κ→∞Eκ(vκ) = 1 2 Z A |∇u|2 = I 0. (6.23)
Actually, the above result was obtained in [9] for minimizers of the Ginzburg-Landau energy Eκ with a fixed Dirichlet boundary data g under the additional assumption that A is simply
connected. However, the simple connectedness of A is used in their proof only for establishing (6.21). Since we proved above (6.21), we may apply Lemma 6.4 to our case.
Note that u is smooth and non constant. Therefore, we may find a compact K ⊂ A and constants c = c(A) > 0, κ′ = κ′(A) > 0 such that
κ2(1− |vκ|2)≥ c in K for κ > κ′(A). (6.24)
Let now κ > κ′(A). Using (6.23), we find that
I0− Eκ(vκ) = X n≥0
(E2n+1κ(v2n+1κ)− E2nκ(v2nκ)). (6.25)
By the minimality of vκ with respect to Eκ, we have
E2n+1κ(v2n+1κ)− E2nκ(v2nκ)≥ E2n+1κ(v2n+1κ)− E2nκ(v2n+1κ) = 3
44
nκ2Z A
(1− |v2n+1κ|2)2. (6.26)
Combining (6.24) and (6.26), we find
E2n+1κ(v2n+1κ)− E2nκ(v2nκ)≥ 3
44
nκ2Z K
Going back to (6.25), we obtain, for κ > κ′(A), that mκ ≤ Eκ(vκ)≤ I0− X n≥0 3c24−n−3κ−2|K| = I0− C κ2. (6.28)
7
Properties of quasi-minimizers
We start by defining the quasi-minimizers we alluded to in the Introduction.
Definition 7.1. A family of quasi-minimizers is a family (uκ)⊂ J , κ ≥ 0, such that :
Eκ(uκ)≤ mκ+
1
eκ (7.1)
and uκ is a minimizer of the Ginzburg-Landau energy with respect to its own boundary condition,
i.e.,
Eκ(uκ)≤ Eκ(v) if tr∂Av = tr∂Auκ. (7.2)
Note that quasi-minimizers always exist. Moreover, any minimizer uκ of mκ, if it exists, is a
quasi-minimizer. It also follows from the maximum principle that
|uκ| ≤ 1 in A. (7.3)
On the other hand, quasi-minimizers satisfy
−∆uκ = κ2uκ(1− |uκ|2) in A (7.4)
and thus they are smooth in A.
Note also that, in view of Proposition 6.1, we have the following uniform bound
Eκ(uκ)≤ 2π + 1 = C. (7.5)
The main tool for determining the asymptotic behavior of the quasi-minimizers is the following Lemma 7.1. ([33]) Set, for z ∈ A, d(z) = dist(z, ∂A). Under the assumptions (7.3)-(7.5), we have |Dlu κ(z)| ≤ Cl dl(z), l ∈ IN, z ∈ A (7.6) and |Dl(1− |uκ|2)(z)| ≤ Cl κ2dl+2(z), l ∈ IN, z ∈ A. (7.7)
Actually, the above estimates were established in [33] when d(z) = 1 ; the general case follows by scaling. Note that these estimates deteriorate when we approach the boundary ; in fact, it is not reasonable to expect uniform estimates up to the boundary.
Corollary 7.1. Let (uκ) be a family of quasi-minimizers. Then, up to subsequences, (uκ)
con-verges, weakly in H1(A) and strongly in Cl
loc(A), l∈ IN, to some u ∈ C∞(A) such that |u| = 1.
Proposition 7.1. Assume that I0 > 2π and let (uκ) be a family of quasi-minimizers. Then there
are constants ακ ∈ S1 such that (ακuκ) converges, weakly in H1(A) and strongly in Clocl (A), l∈ IN,
to 1.
Proof : We start by proving that any possible limit as in Corollary 7.1 is a constant of modulus 1. Indeed, let u be any S1-valued smooth map such that, along some subsequence, (u
κ) converges,
weakly in H1(A) and strongly in Cl
loc(A), l∈ IN, to u. First of all, the proof of Corollary 6.6 implies
that u 6∈ K. Thus u is in some class KD, d
0, . . . , dk with (D, d0, . . . , dk)6= (1, −1, 0, . . . , 0). Since
u∈ H1(A ; S1), Lemma 2.2 implies that D+ k X j=0
dj = 0. Therefore, among the degrees D, d0, . . . , dk,
there are at least two degrees different from the corresponding degrees 1,−1, 0, . . . , 0. Let us assume, for simplicity, that D6= 1 and d0 6= −1, the analysis being similar in the other cases. Fix
some small δ > 0 and let
Γδ ={z ∈ A ; dist(z, ∂Ω) = δ}, Ωδ ={z ∈ A ; dist(z, ∂Ω) < δ}
and respectively
γδ ={z ∈ A ; dist(z, ∂ω0) = δ}, ωδ ={z ∈ A ; dist(z, ∂ω0) < δ}.
We orient Γδ and γδ as boundaries of Ωδ and ωδ, respectively. By Lemma 2.2 applied to u in Ωδ
and in ωδ, we have
deg(u, Γδ) = −D and deg(u, γδ) = −d0. (7.8)
We now argue as in the proof of Proposition 5.1. Assume that uκl ⇀ u as in Corollary 7.1. Then
we have
2π ≥ lim inf mκl ≥ lim inf
Using the C1
loc(A) convergence of uκl to u, we find that
lim inf 1 2 Z Ωδ |∇(uκl− u)| 2 ≥ lim inf 1 2 Z ∂Ω (uκl− u) ∧ ∂(uκl− u) ∂τ . (7.12)
As in the proof of Proposition 5.1, we obtain
lim inf 1 2 Z Ωδ |∇(uκl− u)| 2 ≥ π|D − 1|. (7.13) Similarly, lim inf 1 2 Z ωδ |∇(uκl− u)| 2 ≥ π|d0+ 1|. (7.14)
By combining (7.10), (7.13), (7.14) and the fact that D6= 1, d0 6= −1, we are led to Z
A
|∇u|2 = 0,
so that u is a constant of modulus 1.
We next prove the existence of the family (ακ). Set
βκ = 1 |A| Z A uκ. (7.15)
Since any possible limit u is a constant of modulus 1, it is easy to see that|βκ| → 1 as κ → ∞. If
we set, for sufficiently large κ, ακ =
βκ
|βκ|
, then for this choice the conclusion of the proposition is straightforward.
As a consequence of the above proposition, we obtain that quasi-minimizers have to vanish at least twice in the supercritical case
Lemma 7.2. Assume that A is supercritcal, i.e., that I0 > 2π. Let (uκ) be a family of
quasi-minimizers. Fix some small δ > 0. Then there is some κ′ = κ′(A) such that, for κ > κ′, u κ has
at least a zero at distance < δ from ∂Ω and at least a zero at distance < δ from ∂ω0.
Remark 7.1. We will see in Section 11 that uκ has exactly two zeroes for large κ, but the
argument is rather involved.
Proof of Lemma 7.2 : We reason near ∂Ω, the situation being similar near ∂ω0. Fix some δ > 0
and let Γ ={z ∈ A ; dist(z, ∂Ω) = δ}. Let κ′ be such that, for κ > κ′, we have |α
κuκ − 1| < 1/2
on Γ. Then, for any such κ, we have deg(ακuκ, Γ) = 0, and thus deg(uκ, Γ) = 0. Argue by
